Definition. A mapping (or function f from A to B) of a set A into a set B is a rule that pairs each element of A with exactly one element of a subset of B. The set A is called the domain, and the set of all elements of B (a subset of B) that are paired with an element from A is called the range.
Definition. A mapping f from A to B is one-to-one if each element of the range of f is the image of exactly one element from A.
Theorem 3.0. The set of transformations of a plane is a group under composition.
Proof. The result follows from the following:
The composition of two transformations of a plane is a transformation (Exercise 3.4).
The inverse of a transformation is a transformation (Exercise 3.5).
The identity function is a transformation and composition of functions is associative (Exercise 3.5).//
Exercise 3.2. Which of the following mappings are transformations? Justify.
a. such that .
b. such that .
c. such that .
d. such that .
e. Let P be a point in a plane S. Define by f(P) = P and for any point , f(Q) is the midpoint of .
Exercise 3.3. Let and be transformations defined respectively by f(x,y) = (x 4, y + 1) and g(x,y) = (x + 2, y + 3).
a. Find the composition .
b. Find the composition .
c. Find the inverse of f, f 1.
d. Find the inverse of g, g 1.
Exercise 3.4. Prove the composition of two transformations of a plane is a transformation of the plane.
Exercise 3.5. (a) Prove the identity function is a transformation. (b) Prove the inverse of a transformation of a plane is a transformation of the plane. (c) Prove the composition of functions is associative.
© Copyright 2005, 2006 - Timothy Peil